Following Refs. [1]}, [2]}, [3]}, [4]}, [5]}, we describe the non-equilibrium properties of the MTJ with a single-orbital tight-binding (TB) model in combination with quantum transport simulations. The system is composed of two semi-infinite magnetic leads coupled to an insulating spacer via spin-independent hopping matrices \(H_{I,T(B)} = t\sigma _0\) , where \(\sigma _0 = I_{2\times 2}\)  is the 2\(\times \) 2 identity matrix in spin space and \(t\)  is the associated hopping parameter. The Hamiltonian of the coupled system has the block tridiagonal form
\(H = \left(\begin{array}{ccc}H_T & H_{T,I} & 0 \\H_{T,I}^{\dagger } & H_{I} & H_{I,B} \\0 & H_{I,B}^{\dagger } & H_{B} \\\end{array}\right),\) 
